Uniform Distribution (Continuous) - MATLAB & Simulink (2024)

Overview

The uniform distribution (also called the rectangular distribution) is a two-parameter family of curves that is notable because it has a constant probability distribution function (pdf) between its two bounding parameters. This distribution is appropriate for representing the distribution of round-off errors in values tabulated to a particular number of decimal places. The uniform distribution is used in random number generating techniques such as the inversion method.

Statistics and Machine Learning Toolbox™ offers several ways to work with the uniform distribution.

Parameters

The uniform distribution uses the following parameters.

The standard uniform distribution has a = 0 and b = 1.

Probability Density Function

The pdf of the uniform distribution is

The pdf is constant between a and b.

For an example, see Compute Continuous Uniform Distribution pdf.

Cumulative Distribution Function

The cumulative distribution function (cdf) of the uniform distribution is

The result p is the probability that a single observation from a uniform distribution with parameters a and b falls in the interval [ax].

For an example, see Compute Continuous Uniform Distribution cdf.

Descriptive Statistics

The mean of the uniform distribution is $$\mu =\frac{1}{2}\left(a+b\right)$$.

The variance of the uniform distribution is $${\sigma }^2=\frac{1}{12}{\left(b-a\right)}^2$$.

Random Number Generation

You can use the standard uniform distribution to generate random numbers for any other continuous distribution by the inversion method. The inversion method relies on the principle that continuous cumulative distribution functions (cdfs) range uniformly over the open interval (0, 1). If u is a uniform random number on (0, 1), then x = F^–1(u) generates a random number x from the continuous distribution with the specified cdf F.

For an example, see Generate Random Numbers Using Uniform Distribution Inversion.

Examples

pd1 = makedist('Uniform'); % Standard uniform distributionpd2 = makedist('Uniform','lower',-2,'upper',2); % Uniform distribution with a = -2 and b = 2pd3 = makedist('Uniform','lower',-2,'upper',1); % Uniform distribution with a = -2 and b = 1

x = -3:.01:3;pdf1 = pdf(pd1,x);pdf2 = pdf(pd2,x);pdf3 = pdf(pd3,x);

figure;plot(x,pdf1,'r','LineWidth',2); hold on;plot(x,pdf2,'k:','LineWidth',2);plot(x,pdf3,'b-.','LineWidth',2);legend({'a = 0, b = 1','a = -2, b = 2','a = -2, b = 1'},'Location','northwest');xlabel('Observation')ylabel('Probability Density')hold off;

pd1 = makedist('Uniform'); % Standard uniform distributionpd2 = makedist('Uniform','lower',-2,'upper',2); % Uniform distribution with a = -2 and b = 2pd3 = makedist('Uniform','lower',-2,'upper',1); % Uniform distribution with a = -2 and b = 1

x = -3:.01:3;cdf1 = cdf(pd1,x);cdf2 = cdf(pd2,x);cdf3 = cdf(pd3,x);

figure;plot(x,cdf1,'r','LineWidth',2); hold on;plot(x,cdf2,'k:','LineWidth',2);plot(x,cdf3,'b-.','LineWidth',2);legend({'a = 0, b = 1','a = -2, b = 2','a = -2, b = 1'},'Location','NW');xlabel('Observation')ylabel('Cumulative Probability')hold off;

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